Please help (simplify without using the calculator)

[Ojisan]

Hi there, would there be anyone that could help me with the following
i been on this for long and cant get it

it says ( simplify without using the calculator )

5³ⁿ⁺¹ - 5^{3n-1
____________________}
5³ⁿ⁺¹

also this one which says ( Solve for x: )

2^x+1 + 2^x+2 = 192


Many thanks

 

[pappus]

Hi there, would there be anyone that could help me with the following
i been on this for long and cant get it

it says ( simplify without using the calculator )

5³ⁿ⁺¹ - 5^{3n-1
____________________}
5³ⁿ⁺¹

also this one which says ( Solve for x: )

2^x+1 + 2^x+2 = 192


Many thanks

\(\displaystyle \displaystyle{\frac{5^{3n+1}-5^{3n-1}}{5^{3n+1}}= \frac{5^{3n+1}}{5^{3n+1}}-\frac{5^{3n-1}}{5^{3n+1}}}\)

The 1st summand is 1, with the 2nd summand use \(\displaystyle \displaystyle{\frac{b^x}{b^y} = b^{x-y}}\)

\(\displaystyle \displaystyle{2^{x+1} + 2^{x+2} = 192~\implies~2 \cdot 2^x + 4 \cdot 2^x = 192}\)

Collect like terms, divide through by 6 and re-write 32 as a power of 2. Two powers with the same base are equal if the exponents are equal too.

 

[Ojisan]

\(\displaystyle \displaystyle{\frac{5^{3n+1}-5^{3n-1}}{5^{3n+1}}= \frac{5^{3n+1}}{5^{3n+1}}-\frac{5^{3n-1}}{5^{3n+1}}}\)

The 1st summand is 1, with the 2nd summand use \(\displaystyle \displaystyle{\frac{b^x}{b^y} = b^{x-y}}\)

\(\displaystyle \displaystyle{2^{x+1} + 2^{x+2} = 192~\implies~2 \cdot 2^x + 4 \cdot 2^x = 192}\)

Collect like terms, divide through by 6 and re-write 32 as a power of 2. Two powers with the same base are equal if the exponents are equal too.

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Thanks for the help